We study the conjugate of the maximum,
f
∨
g
f \vee g
, of
f
f
and
g
g
when
f
f
and
g
g
are proper convex lower semicontinuous functions on a Banach space
E
E
. We show that
(
f
∨
g
)
∗
∗
=
f
∗
∗
∨
g
∗
∗
(f \vee g)^{**} = f^{**} \vee g^{**}
on the bidual,
E
∗
∗
E^{**}
, of
E
E
provided that
f
f
and
g
g
satisfy the Attouch-Brézis constraint qualification, and we also derive formulae for
(
f
∨
g
)
∗
(f \vee g)^{*}
and for the “preconjugate” of
f
∗
∨
g
∗
f^{*}\vee g^{*}
.